12 research outputs found
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Polarizing Double Negation Translations
Double-negation translations are used to encode and decode classical proofs
in intuitionistic logic. We show that, in the cut-free fragment, we can
simplify the translations and introduce fewer negations. To achieve this, we
consider the polarization of the formul{\ae}{} and adapt those translation to
the different connectives and quantifiers. We show that the embedding results
still hold, using a customized version of the focused classical sequent
calculus. We also prove the latter equivalent to more usual versions of the
sequent calculus. This polarization process allows lighter embeddings, and
sheds some light on the relationship between intuitionistic and classical
connectives
Semantic A-translation and Super-consistency entail Classical Cut Elimination
We show that if a theory R defined by a rewrite system is super-consistent,
the classical sequent calculus modulo R enjoys the cut elimination property,
which was an open question. For such theories it was already known that proofs
strongly normalize in natural deduction modulo R, and that cut elimination
holds in the intuitionistic sequent calculus modulo R. We first define a
syntactic and a semantic version of Friedman's A-translation, showing that it
preserves the structure of pseudo-Heyting algebra, our semantic framework. Then
we relate the interpretation of a theory in the A-translated algebra and its
A-translation in the original algebra. This allows to show the stability of the
super-consistency criterion and the cut elimination theorem
Resolution in Solving Graph Problems
International audienceResolution is a proof-search method for proving unsatisfia-bility problems. Various refinements have been proposed to improve the efficiency of this method. However, when we try to prove some graph properties, it seems that none of the refinements have an efficiency comparable with traditional graph traversal algorithms. In this paper we propose a way of encoding some graph problems as resolution. We define a selection function and a new subsumption rule to avoid redundancies while solving such problems
Superposition with Structural Induction
International audienceSuperposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved. We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging AVATAR [1]. Lemmas obtained by generalization or theory exploration can be introduced during search, used, and proved, all in the same search space. We describe an implementation and present some promising results
Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo
In deduction modulo, a theory is not represented by a set of axioms but by a
congruence on propositions modulo which the inference rules of standard
deductive systems---such as for instance natural deduction---are applied.
Therefore, the reasoning that is intrinsic of the theory does not appear in the
length of proofs. In general, the congruence is defined through a rewrite
system over terms and propositions. We define a rigorous framework to study
proof lengths in deduction modulo, where the congruence must be computed in
polynomial time. We show that even very simple rewrite systems lead to
arbitrary proof-length speed-ups in deduction modulo, compared to using axioms.
As higher-order logic can be encoded as a first-order theory in deduction
modulo, we also study how to reinterpret, thanks to deduction modulo, the
speed-ups between higher-order and first-order arithmetics that were stated by
G\"odel. We define a first-order rewrite system with a congruence decidable in
polynomial time such that proofs of higher-order arithmetic can be linearly
translated into first-order arithmetic modulo that system. We also present the
whole higher-order arithmetic as a first-order system without resorting to any
axiom, where proofs have the same length as in the axiomatic presentation
CTL Model Checking in Deduction Modulo
International audienceIn this paper we give an overview of proof-search method for CTL model checking based on Deduction Modulo. Deduction Modulo is a reformulation of Predicate Logic where some axioms—possibly all—are replaced by rewrite rules. The focus of this paper is to give an encoding of temporal properties expressed in CTL, by translating the logical equivalence between temporal operators into rewrite rules. This way, the proof-search algorithms designed for Deduction Modulo, such as Resolution Modulo or Tableaux Modulo, can be used in verifying temporal properties of finite transition systems. An experimental evaluation using Resolution Modulo is presented
Linking Focusing and Resolution with Selection
International audienceFocusing and selection are techniques that shrink the proof-search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atomic formula can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory and the related framework called superdeduction; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof-search space
Constrained narrowing for conditional equational theories modulo axioms
For an unconditional equational theory (Sigma, E) whose oriented equations (E) over arrow are confluent and terminating, narrowing provides an E-unification algorithm. This has been generalized by various authors in two directions: (i) by considering unconditional equational theories (Sigma, E boolean OR B) where the (E) over arrow are confluent, terminating and coherent modulo axioms B, and (ii) by considering conditional equational theories. Narrowing for a conditional theory (Sigma, E boolean OR B) has also been studied, but much less and with various restrictions. In this paper we extend these prior results by allowing conditional equations with extra variables in their conditions, provided the corresponding rewrite rules (E) over arrow are confluent, strictly coherent, operationally terminating modulo B and satisfy a natural determinism condition allowing incremental computation of matching substitutions for their extra variables. We also generalize the type structure of the types and operations in Sigma to be order-sorted. The narrowing method we propose, called constrained narrowing, treats conditions as constraints whose solution is postponed. This can greatly reduce the search space of narrowing and allows notions such as constrained variant and constrained unifier that can cover symbolically possibly infinite sets of actual variants and unifiers. It also supports a hierarchical method of solving constraints. We give an inference system for hierarchical constrained narrowing modulo B and prove its soundness and completeness. (C) 2015 Elsevier B.V. All rights reserved.We thank the anonymous referees for their constructive criticism and their very detailed and helpful suggestions for improving an earlier version of this work. We also thank Luis Aguirre for kindly giving us additional suggestions to improve the text. This work has been partially supported by NSF Grant CNS 13-19109 and by the EU (FEDER) and the Spanish MINECO under grant TIN 2013-45732-C4-1-P, and by Generalitat Valenciana PROMETEOII/2015/013.Cholewa, A.; Escobar Román, S.; Meseguer, J. (2015). Constrained narrowing for conditional equational theories modulo axioms. Science of Computer Programming. 112:24-57. https://doi.org/10.1016/j.scico.2015.06.001S245711
Le model Checking et la émonstration de théorèmes
Model checking is a technique for automatically verifying correctness properties of finite systems. Normally, model checking tools enjoy two remarkable features: they are fully automatic and a counterexample will be produced if the system fails to satisfy the property. Deduction Modulo is a reformulation of Predicate Logic where some axioms—possibly all—are replaced by rewrite rules. The focus of this dissertation is to give anencoding of temporal properties expressed in CTL as first-order formulas, by translating the logical equivalence between temporal operators into rewrite rules. This way, proof-search algorithms designed for Deduction Modulo, such as Resolution Modulo or Tableaux Modulo, can be used to verify temporal properties of finite transition systems.To achieve the aim of solving model checking problems with an off-the-shelf automated theorem prover, three works are included in this dissertation. First, we address the graph traversal problems in model checking with automated theorem provers. As a preparationwork, we propose a way of encoding a graph as a formula such that the traversal of the graph corresponds to resolution steps. Then we present the way of translating model checking problems as proving first-order formulas in Deduction Modulo. The soundness and completeness of our method shows that solving CTL model checking problems with automated theorem provers is feasible. At last, based on the theoreticalbasis in the second work, we propose a symbolic model checking method. This method is implemented in iProver Modulo, which is a first-order theorem prover uses Polarized Resolution Modulo.Le model checking est une technique de vérification automatique de propriétés de correction de systèmes finis. Normalement, les outils de model checking ont deux caractéristiques remarquables: ils sont automatisés et ils produisent un contre-exemple si le systéme ne satisfait pas la propriété. La Déduction Modulo est une reformulation de la logique des prédicats où certains axiomes—possiblement tous—sont remplacés par des régles de réécriture. Le but de cette dissertation est de donner un encodage de propriétés temporelles exprimées en CTL en des formules du premier ordre, en exprimant l’équivalence logique entre les opérateurs temporels avec des règles de réécriture. De cette manière, les algorithmes de recherche de preuve conçus pour la Déduction Modulo, tels que la Résolution Modulo ou les Tableaux Modulo, peuvent être utilisés pour vérifierdes propriétés temporelles de systèmes de transition finis.Afin d’accomplir le but de résoudre des problèmes de model checking avec un prouveur automatique quelconque, trois travaux sont inclus dans cette dissertation. Premièrement, nous abordons le problème de parcours de graphes en model checking avec des prouveurs automatiques. Nous proposons une façon d’encoder un graphe en tant que formule de manière à ce que le parcours du graphe correspond aux etapes de résolution. Nous présentons ensuite comment formuler les problèmes de model checking comme des formules du premier ordre en Déduction Modulo. La correction et la complétude de notre méthode montre que résoudre des problèmes de model checking CTL avec des prouveursautomatiques est faisable. Enfin, en nous appuyant sur la base théorique du deuxième travail, nous proposons une méthode de model checking symbolique. Cette méthode est implantée dans iProver Modulo, qui est un prouveur automatique du premier ordre qui utilise la Résolution Modulo Polarisée